Name: kaplan-meier survival curve basics
Uploaded: 2025-01-09T15:14:16.000Z
Duration: 1 min 18 s
Channel: JoVE

Рассмотрим график кумулятивной вероятности смерти, представленный в виде возраста по оси X, в зависимости от доли умерших по оси Y за определенный год.
Это можно выразить в виде уравнения, где кумулятивная функция распределения F(t) представляет собой отношение числа людей, умерших к моменту времени t, к общему числу наблюдаемых.
Поскольку все члены популяции не наблюдаются до смерти, эта кривая не может оценить выживаемость.
Таким образом, функция выживания или кривая выживания — S(t) — это доля или процент людей, живущих до момента t или позже. Выражается он следующим образом.
Затем кривая выживаемости строится с использованием возраста и процента выживших людей.
Существуют различные виды моделей выживания. Экспоненциальная модель выживания характеризует постоянную опасность во времени, что означает, что риск наступления события не зависит от времени.
Модель выживания Вейбулла может быть использована в различных ситуациях, когда уровень опасности монотонно увеличивается или уменьшается с течением времени.
Логарифмическая нормальная и логарифмическая логистические модели могут использоваться в сценариях, когда уровень опасности не является монотонным.

Кривые выживаемости

Survival Curves

Кривые выживаемости — это графические представления, которые отображают опыт выживания популяции с течением времени, предлагая интуитивно понятный способ отслеживания доли людей, которые остаются без событий в каждый момент времени. Эти кривые широко используются в таких областях, как медицина, здравоохранение и инженерия надежности, для визуализации и сравнения вероятностей выживания в различных группах или условиях. 
 Оценка Каплана-Майера — наиболее распространенный метод построения кривых выживаемости. Этот непараметрический подход генерирует ступенчатую функцию, где кривая падает каждый раз, когда происходит событие (такое как смерть, рецидив заболевания или механическая поломка). Горизонтальные сегменты между падениями указывают на периоды стабильности, в течение которых не происходит никаких событий. Ось x кривой представляет время, а ось y показывает вероятность выживания в диапазоне от 0 до 1. Кривые выживаемости дают несколько ключевых идей: 

 <ol> 
 	<li>Вероятность выживаемости с течением времени: Кривая иллюстрирует вероятность того, что люди выживут после определенных моментов времени. Например, если кривая выживаемости для группы лечения остается выше, чем у контрольной группы, это указывает на эффективность лечения в продлении жизни или отсрочке события.</li> 
 	<li>Медианное время выживания: Медианное время выживания — это точка, в которой вероятность выживания падает до 0,5, указывая время, к которому, как ожидается, половина когорты испытает событие. Эта метрика особенно важна в клинических исследованиях как ориентир для эффективности лечения.</li> 
 	<li>Сравнение групп: Кривые выживаемости — это мощный инструмент для сравнения выживаемости разных групп, например, пациенты, проходящие различные виды лечения, или системы, подвергающиеся различным стрессовым воздействиям. Статистические тесты, такие как логранговый тест, часто используются вместе с кривыми выживаемости, чтобы определить, являются ли наблюдаемые различия между группами статистически значимыми.</li> 
 </ol> 
 Например, в клиническом исследовании, сравнивающем два метода лечения рака, кривые выживаемости могут показать, какой метод лечения обеспечивает лучшие результаты выживания. Кривая, которая снижается более плавно, указывает на группу с лучшими вероятностями выживания. Аналогично, в области надежности кривые выживаемости используются для оценки срока службы компонентов или систем, что позволяет эффективно планировать техническое обслуживание и анализировать отказы. 
 Предоставляя четкое и доступное визуальное представление сложных данных о времени до события, кривые выживаемости играют важную роль в анализе данных. Их способность суммировать вероятности выживаемости, определять ключевые показатели, такие как медианное время выживаемости, и облегчать групповые сравнения, делает их незаменимыми в различных областях применения.

Рассмотрим график кумулятивной вероятности смерти, представленный в виде возраста по оси X, в зависимости от доли умерших по оси Y за определенный год.
Этот...

Education

JoVE Core

Statistics

Статистика

Unit 1

Блок 1

Survival Analysis

Анализ выживаемости

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

kaplan-meier survival curve basics

Kaplan-Meier survival curves show how the chance of staying event-free changes over time. They are used to track survival in a population and to compare groups in medicine, public health, and reliability engineering.

The Kaplan-Meier estimator is the most common way to build these curves. It is a non-parametric method, which means it does not assume a fixed pattern for the data. The result is a stepwise line that drops each time an event happens, such as death, disease recurrence, or mechanical failure.

The flat parts of the curve matter too. They show time periods when no events occur. On the graph, the x-axis shows time and the y-axis shows survival probability, which ranges from 0 to 1.

These curves help show survival probability over time. If one treatment group stays higher than a control group, it suggests better survival or a longer delay before the event. The median survival time is the point where the curve reaches 0.5, meaning half the group has had the event by that time.

Survival curves are also useful for group comparisons. Researchers may compare patients receiving different treatments or systems under different stress conditions. A log-rank test is often used to check whether the difference between curves is statistically significant. In a cancer trial, for example, a curve that falls more slowly can show better survival outcomes.

In reliability engineering, survival curves help estimate how long a component or system may last. That information supports maintenance planning and failure analysis. Because they turn time-to-event data into a clear graph, survival curves are an important tool for data analysis.

Kaplan-Meier Survival Curve Basics

introduction to survival analysis

Time-to-Event Data and Censoring

Survival analysis is a statistical method for time-to-event data. The event can be death, disease relapse, system failure, or recovery. It helps estimate survival time, which is the time until a specific event occurs, and it also shows what factors may affect that time. A key idea in survival analysis is censoring. Censoring happens when the event of interest has not been observed for some people during the study period. Because the data are incomplete in those cases, survival analysis uses...

life tables

Life Tables and Survival Patterns

A life table is a statistical tool that shows mortality and survival patterns in a population. It organizes data across age intervals so students can see how likely a group is to survive or die at each stage of life. The table gives a clear snapshot of population dynamics over time. Life tables are used in demography, public health, actuarial science, and ecology. They help scientists study life expectancy, plan health interventions, estimate insurance risk, and compare how species survive in...

survival curves

Kaplan-Meier survival curves show how the chance of staying event-free changes over time. They are used to track survival in a population and to compare groups in medicine, public health, and reliability engineering. The Kaplan-Meier estimator is the most common way to build these curves. It is a non-parametric method, which means it does not assume a fixed pattern for the data. The result is a stepwise line that drops each time an event happens, such as death, disease recurrence, or...

actuarial approach

Life Table Method for Survival Rates

The life table method, also called the actuarial approach, is used to estimate survival rates in clinical and population studies. It began in life insurance, where risk over time must be measured carefully. In health research, it gives a clearer picture of survival when some participants are lost to follow-up or die from causes unrelated to the study. This method works by dividing the study period into set intervals, usually one year. That makes it useful for tracking outcomes over time,...

kaplan-meier approach

Survival Curves with Kaplan-Meier Data

The Kaplan-Meier estimator is a non-parametric method for estimating a survival function from time-to-event data. It is commonly used in medical research to track how many patients survive for a set time after treatment. It is also important in clinical trials, epidemiological studies, and reliability engineering. This method helps researchers study survival probabilities over time, even when some data are incomplete. That incomplete data is called censored data, which means the exact event...

assumptions of survival analysis

Survival Analysis Assumptions in Studies

Survival analysis studies the time until an event happens. The event may be death in a living organism or failure in a mechanical system. These models are used in medicine, biology, engineering, and public health to study time-to-event data. Careful study design matters because the method depends on several key assumptions. One common feature of survival times is positive skewness. This means events often happen early, while fewer cases occur later. Survival data also often include censoring,...

comparing the survival analysis of two or more groups

comparing-the-survival-analysis-of-two-or-more-groups

Kaplan-Meier Curves and Hazard Ratios

Kaplan-Meier curves and hazard ratios help researchers compare survival time between groups. Survival analysis looks at the time until an event happens, such as death, disease recurrence, or recovery. It is a key tool in medical research. A major strength of survival analysis is that it can handle censored data. Censored data means the event has not happened for some participants by the end of the study, or it has not been observed. This makes survival analysis different from standard...

the mantel-cox log-rank test

Log-Rank Test for Comparing Survival Curves

The Mantel-Cox log-rank test compares survival curves from two groups. It checks whether the groups have a statistically significant difference in survival time. This makes it useful when the timing of an event matters, such as death or disease recurrence. The log-rank test is non-parametric. That means it does not assume a specific distribution for the survival data. It is also useful in medical research, where it often helps compare a new treatment with a control or standard treatment in...

applications of life tables

Life Tables in Public Policy and Science

Life tables are used in many fields to study mortality and survival rates. They give a numerical way to examine how long people or living things are likely to survive. This makes them useful for public health, business planning, and scientific research. In demography and public health, life tables help experts study population changes and predict future trends. They can show patterns such as an aging population or the effect of a health policy. In Nigeria, health officials used survival rates...

cancer survival analysis

Cancer Survival Curves and Risk Factors

Cancer survival analysis measures how long it takes from diagnosis or the start of treatment to a specific outcome, such as remission or death. It helps researchers judge treatment effectiveness and understand what affects patient outcomes. These results also support clinical decisions and prognostic evaluations. Survival data are often skewed and not normally distributed. They also include censored cases, where the full survival time is unknown because the study ends, a patient is lost to...

hazard rate

Modeling Event Risk Over Time

Hazard rate is a statistical measure that describes how event risk changes over time. It is also called the hazard function or failure rate. In survival analysis, it shows the instantaneous rate at which an event occurs, but only after the subject has not had the event yet. From a probability view, hazard rate gives the chance that a subject will have the event in a very small time interval. That chance is conditional on surviving to the start of that interval. From a frequency view, it can...

hazard ratio

Hazard Ratio in Clinical Trials

The hazard ratio is a way to compare event risk in clinical trials. It is often used to study events such as death or disease recurrence in two groups over time. The two groups are usually a treatment group and a control group. A hazard ratio compares hazard rates, which are the instantaneous risk of an event happening at a given moment. It shows how the risk in one group relates to the risk in the other group. In a cancer drug study, for example, the hazard ratio can show how a new treatment...

truncation in survival analysis

Left and Right Truncation in Survival Data

Truncation in survival analysis describes data loss caused by the timing of an event or by when a subject can enter a study. It can happen in two forms: left truncation and right truncation. In both cases, some individuals are left out of the dataset before analysis begins. Left truncation happens when people who had the event before a set time are not included in the study. This often occurs with delayed entry, which means a person is only observed if they survive until the entry point. For...

censoring survival data

Types of Censoring in Survival Analysis

Survival analysis studies time-to-event data, which tracks how long it takes for an outcome to happen. It is used in medicine, engineering, and social science. A common problem in this type of data is censoring, which means the full event time is not observed for every person or item. Censoring can happen for several reasons and in different patterns. The most common kind is right censoring. This happens when the event has not occurred by the end of the study or when a participant is lost to...

survival tree

Survival Tree in Time-to-Event Analysis

A survival tree is a non-parametric tool in survival analysis that links predictor variables, or covariates, to survival time. Survival time is the time until an event of interest occurs. It is useful when some data are censored, which means the event has not happened for some people by the end of the study or the exact event time is unknown. Building a survival tree starts with a dataset that includes covariates, survival time, and a censoring indicator for each subject. The data must be...

parametric survival analysis: weibull and exponential methods

parametric-survival-analysis-weibull-and-exponential-methods

Modeling Survival Time with Weibull and Exponential

Parametric survival analysis models survival time by assuming a specific probability distribution for when an event occurs. The Weibull and exponential distributions are two common choices because they are flexible and relatively straightforward to use. The Weibull distribution is a flexible model in parametric survival analysis. It can describe both increasing and decreasing hazard rates, depending on its shape parameter beta. When beta is greater than 1, the hazard rate rises over time. That...

key terms and concepts

Learn how Kaplan-Meier survival curves track event-free survival over time, show median survival time, and compare groups in medicine and engineering.

Kaplan-Meier Survival Curve Basics | JoVE K12

statistics

JoVE Core Statistics introduces the concepts and application of the collection, analysis, presentation, and interpretation of data to students. These animated lessons provide an understanding of statistics beneficial to students from a wide range of disciplines. Additionally, the scientist-in-action videos demonstrate the application of related concepts in classical and original research experiments performed in today's laboratories worldwide

Kaplan-Meier Survival Curve Basics

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